Polynomials

A polynomial is an expression built from terms that are a number times a variable raised to a whole-number power, added or subtracted together. For example, 3x² − 5x + 2 is a polynomial with three terms. The highest power of the variable is the polynomial's degree, so 3x² − 5x + 2 has degree 2. Linear, quadratic and cubic expressions are all polynomials.

What is a polynomial?

The word means "many terms". Take 4x³ − 7x + 9:

• Terms: 4x³, −7x and 9, the parts joined by + and −.
• Coefficients: the numbers in front of the variable, 4 and −7.
• Constant: the term with no variable, 9.
• Variable: x.
• Exponents: the powers, which must be whole numbers 0, 1, 2, 3 and so on.

What is not a polynomial

The rule about whole-number powers rules out a lot. A variable can't be in a denominator, under a square root, or in an exponent.

Degree of a polynomial and its name

Polynomials are named by their degree and, for short ones, by how many terms they have.

With more than one variable, add the powers in each term and take the biggest: 3x²y³ + xy has degree 5. Try identifying some degrees yourself, then press Mark to check in the course.

Standard form and leading coefficient

A polynomial is in standard form when its terms run from the highest power to the lowest: 5 + 2x³ − x becomes 2x³ − x + 5. The first term is the leading term, and its number, here 2, is the leading coefficient. The leading term decides what the graph does at its far ends.

Adding and subtracting polynomials

Collect like terms: terms with the same variable and the same power.

To subtract, change the sign of every term in the second bracket, then add:

Multiplying polynomials

Each term of one bracket multiplies each term of the other. Then collect like terms:

When you multiply polynomials, the degrees add: a degree 1 times a degree 1 gives degree 2. In England, expanding products of two or more binomials is GCSE Higher content in AQA's specification.

Dividing and factoring polynomials

Take f(x) = x³ − 2x² − 5x + 6. The factor theorem says that if f(a) = 0, then (x − a) is a factor. Try small numbers: f(1) = 1 − 2 − 5 + 6 = 0, so (x − 1) is a factor.

Divide f(x) by (x − 1), using long division or synthetic division, to get x² − x − 6. That quadratic factors too:

The polynomial division calculator shows every step of the division, and the factoring calculator factors quadratics with working. If a division doesn't come out exactly, the remainder theorem tells you the remainder in advance: dividing f(x) by (x − a) leaves f(a).

Roots and graphs of polynomials

The roots (or zeros) of a polynomial are the x values that make it zero, which are where its graph crosses the x-axis. From the factors above, x³ − 2x² − 5x + 6 has roots at x = 1, 3 and −2.

Here x is the input and y is the value of the polynomial. The curve crosses the x-axis at the three roots and the y-axis at 6, the constant term. A polynomial of degree n has at most n roots and at most n − 1 turning points, so this cubic has two turning points.

End behaviour

Far to the left and right, only the leading term matters. Its degree and sign tell you which way the ends of the graph point:

Polynomial and rational functions are Unit 1 of AP Precalculus, worth 30% to 40% of the multiple-choice section. The Graphing Functions course has a polynomial shape explorer where you change the degree and coefficients and watch the curve respond.

Common mistakes

• Reading the degree from the first term when the polynomial isn't in standard form. In 4 + x − x³ the degree is 3.
• Only changing the first sign when subtracting a bracket. Every term in the second bracket changes sign.
• Missing terms when multiplying. Two terms times three terms makes six products before you collect.
• Calling 1/x or √x a polynomial term. Negative and fractional powers aren't allowed.
• Adding unlike terms, such as 3x² + 2x = 5x³. They stay separate.

To practise polynomials among other topics in a short timed paper, try today's AP Precalculus paper. In the UK, the same practice opens as A-level Maths: Algebra & Functions.

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Maths · AlgebraRemainder theorem: statement, proof and worked examplesThe remainder theorem explained: why f(a) is the remainder, dividing by ax − b, finding unknown coefficients, and its link to the factor theorem.